Hey, guys. We’re starting an RPG in the Game Room thread, and I have what I imagine is a fairly simple question regarding dice probability.
In the game, actions are determined by simply: roll a die (6, 8, or 10 sided) one time and the result dictates your success.
Several abilities modify that rule: roll TWICE and pick the better result.
Statistically speaking, how big an improvement is this? How much does it affect the average roll?
Thanks, Dopers!
For a standard six-sided die, it’s a pretty big improvement. Here’s the new frequency table:
1 1/36
2 3/36
3 5/36
4 7/36
5 9/36
6 11/36
In general, the probability of rolling k on an n-sided die under this scheme will be (2k - 1)/n[sup]2[/sup]. It’s very easy to prove with images, but hard to explain in words.
Rephrasing ultrafilter’s result:
Probability with 1 Roll allowed P with 2 Rolls P with 3 Rolls
1 or less 1/6 (1/6)[SUP]2[/SUP] (1/6)[SUP]3[/SUP]
2 or less 2/6 (2/6)[SUP]2[/SUP] (2/6)[SUP]3[/SUP]
3 or less 3/6 (3/6)[SUP]2[/SUP] (3/6)[SUP]3[/SUP]
4 or less 4/6 (4/6)[SUP]2[/SUP] (4/6)[SUP]3[/SUP]
5 or less 5/6 (5/6)[SUP]2[/SUP] (5/6)[SUP]3[/SUP]
6 or less 6/6 (6/6)[SUP]2[/SUP] (6/6)[SUP]3[/SUP]
For a k-sided die, the average best of 2 rolls is
(4k[SUP]2[/SUP] + 3k - 1) / ( 6k)
For those who don’t want to spend a few seconds tapping at a calculator, the result of the above two posts is an average “score” of 4.47, compared with an average of 3.5 for a single roll of a six-sided die. So it basically ups the average by nearly a full “point”.
I don’t see a definition of “better” in the OP. Is better always higher, or lower, or odd/even, or what?
The simple answer is that you have doubled the odds of getting a desirable number, whatever “desirable” means. To make any more meaningful calculation you’d have to define the categories of outcome, as in craps.
Correct, if we replace “doubled” with “improved”. Septimus made the assumption that greater is better for illustrative purposes; the table would just be upside down if lesser were better. If there’s some other rule for better we might have to fall back on ultrafilter’s table.
In any case, the odds aren’t doubled, as should be obvious from the tables and forumlas above.
Even further, you’ve got to consider if 4, say, is better or equivalent to a 6. If it’s success or failure, it’d be different than if it were for hitpoint subtraction.
I’ll confess to a certain rustiness with stat, but given the simple statement of the problem, how could the odds not be doubled? The terms, such as number of die sides, are irrelevant. The order of the rolls is irrelevant (and that’s where I think the above calculations are going awry - assuming that the second roll must improve on the first). There is no definition of which outcome is better, only that one outcome will be superior to the other.
Instead of one roll, the player gets two, and is allowed to choose either result independently. The odds of a “better” outcome are doubled, less only the 1/n chance of rolling the same number.
It’s not always clear what ‘doubling’ odds means, but I think you can reasonably say re-rolling comes close to doubling the odds.
If you need a six, re-rolling takes the odds of success from 6/36 to 11/36, and if you need a five (or six), re-rolling takes the odds from 12/36 to 20/36.
Down to if you just need to not get a one (i.e. two or higher), odds go from 30/36 to 35/36. Which is not doubling the chance of success, but is reducing the chance of failure by a factor of six! (Again, ‘doubling’ odds is a funny thing).
The odds of getting an even number on a single roll is 0.5. The odds of getting an even number on your choice of two rolls is not 1.0.
For low-probability events, taking two shots gives you approximately double the probability. But it’s only approximate, and it’s a bad approximation if p isn’t sufficiently small compared with 1.
The last sentence is exactly the answer to your question on why the odds aren’t doubled. With a n sided die 1/n of the time you get the same number twice. Assuming higher is better, then of the remaining (n-1)/n of the time you get a better number on the second roll half of the time on average. Those are the only times a second roll improves things so you get a better roll with two chances than one chance (n-1)/(2n) of the time over all. For a standard die, two rolls will give you a better single outcome
5/12 of the time over all.
Okay, I understand that in any case the odds are slightly less than doubled, because of the chances of rolling the same number twice. I maintain that all other calculations in this thread are meaningless, because the OP has not established what an “improved” roll is or even that it’s as simple as getting a higher number. RPGs tend to be a bit complex in evaluating rolls, so, as in craps, the categories of better and worser rolls would have to be defined for any calculations beyond that first elementary one to have meaning.
There is also a continuing notion that the second roll must improve on the first, which is contrary to the stated problem (roll twice, pick whichever roll is best). There’s obviously a real-world limitation there, in that if you roll the optimal value first you won’t bother with a second roll… but I’d prefer to see this all resolved with the real conditions rather than assumptions and what-ifs.
The calculations do assume that higher rolls are better. If that’s not the case, then we need to have a definition of what better means.
And, in case anyone is really interested in extending the question to n-sided dies*, one re-roll will on average boost** the score by just under 1/6 of n. [(n^2-1)/6n]
So for a six-sided die, since the re-rolling advantage is just under 1.0, if you have a choice of getting +1 to your roll, or getting a re-roll, you’re better off taking a +1 than a re-roll.
But for a 20-sided, the advantage of re-rolling is about 3.3 points, so you’d want a +4 to make the trade***.
** Let’s just assume high is good, for simplicity in talking about things. Analysis works just as well if low is better.
**assuming you only care about the average total. With pass/fail rolls, it’s a little more complicated: obviously if you only care about not getting a 1, then the plus is always better than the re-roll, and if you need a 22 on a d20, then the plus is also always better. But if you only care about rolling a five or higher, then re-rolling is even better than a +4.